what is a "powerset" with base larger than 2?
A powerset $P(S) $of some set $S$ can be treated as all different possible ways of partition $S$ into 2 ordered pair of disjoint subsets.
And I'm curious what is the equivalence of partition $S$ into more than 2 subsets?
For example, partition the set S into 3 ordered pair of disjoint subsets.
With $|S| \geq 3$, The cardinality should be $3^{|S|}$? And moreover what is the name of arbitrary base powersets?
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$\begingroup$The power set of $S$ is (by the usual definition) the set of subsets of $S$. The fact that each subset can be paired with its complement is an interesting accident. You can think of each such pair as a partition into two blocks, but to recover the power set you have to be able to distinguish the blocks. That means you are looking at ordered partitions, as in the comment from @LeoMosher .
An equivalent definition of the power set is as the set of all functions$$ f: S \to \{0,1\} . $$The proof follows from the fact that each subset is determined by and determines its characteristic function: the function that's $1$ on the subset and $0$ elsewhere.
So perhaps the natural generalization you seek is the set of functions$$ f: S \to \{0,1,2\} . $$These correspond to ordered partitions into three blocks (with a little care taken since two of the blocks can be empty simultaneously).
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